Symmetric-group subset permutation representation

Let $S_n$ act on the $r$-element subsets of $\{1,\dots ,n\}$, with permutation character ${\pi }_r$. If $0\le l\le k\le n/2$, thenbecause orbits of pairs are classified by intersection size $0,\dots ,l$.

For $r\le n/2$, the equivariant map from $(r-1)$-subsets to $r$-subsets that sends each subset to the sum of the $r$-subsets containing it is injective over $\mathbb{C}$. Hence ${\pi }_r-{\pi }_{r-1}$ is the character of a representation.